Material continuum

Of course also the differential balances represent a simplification, based on the idea of a material continuum. TTie continuum is characterized by certain fields of scalar quantities (temperature, concentrations, ) and vector quantities (local velocity of mass flow, diffusion fluxes, — ). Generally, the continuum is heterogeneous the fields are continuous in any of the phases, possibly discontinuous at interfaces. Imagine a distillation or absorption column, or a heterogeneous reactor. 

Let us now consider one of the phases, thus a region in space occupied by a continuous multicomponent mixture. The actual nature (molecular configuration) of the mixture may be little known imagine, e.g., a liquid ionic solution with various degrees of dissociation, solvatation, etc. One then usually assumes that some equilibria (such as ionic equilibria) are installed very rapidly so that the (instantaneous local) thermodynamic state at a point of the mixture can be defined by the temperature, pressure, and mass (or mole) fractions of certain K components C, —, . The Q are some formal chemical species the 

p is mass density and yk th mass fraction, t is time and div the divergence operator v is local mass flow velocity (vector) and jk the it-th molecular diffusion flux vector, added to the term pykV representing the convection of particles Ck by the motion of a material element as a whole. So the instantaneous local change (increase) of the Ck-concentration (mass per unit volume) equals minus the amount that escapes from a volume element (the divergence term) plus the amount produced by chemical reactions. Physically, the balance makes sense if we know how the flux jk depends on the gradients (most simply by Pick s law), and how the rates of possible reactions depend on the local state of the element. If also the latter information is available then the balance takes the form of convective diffusion equation, possibly with chemical reactions. [If we have no information on the reaction rates, the w -terms can be eliminated from Eqs. (C.2) by an algebraic transformation in the same manner as in Chapter 4 indeed, it is sufficient to substitute for W, in (4.3.2) and to define the components of column vector n as follows from (C.2).] Observe finally that we have 

We have of course not considered nuclear reactions, and for the sake of simplicity, nor the presence of electrically charged particles in the balances. So neither electric currents are admitted and let us preclude electromagnetic phenomena at all. Then the differential energy balance reads 

In chemical engineering thermodynamics, for different reasons more familiar than internal energy is another function, viz. the enthalpy. The specific enthalpy (per unit mass) equals 

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For other situations, especially when the material continuum is slowly varying and tiny changes in the laser frequencies in the one photon transition have absolutely no effect on the product ratios, our method allows for control, via the optical induction of resonances, in complete generality. 

MODELING FLOW IN POROUS MATERIALS Continuum-Scale Modeling... 

Eringen, A.C., Suhubi, E.S., Chang, T.S., Dill, E.H. Constitutive equations for simple materials continuum physics. In Eringen, A.C. (ed.) Continuum Mechanics of Single Substance Bodies, vol. II. Academic, New York (1975)... 

Accepting that atoms do not collapse, why then do they not all combine into a single giant molecule, making all of us pieces of the same material continuum, a piece of a single universal entity Here comes to play the Exclusion Rule, which when applied to electrons imposes upon all atoms a specific connectivity that chemists discovered already in the nineteenth century and called it valency, and which in our course is simply the atomic connectivity in the bond construction process. It is the Exclusion Rule that underlies the division of matter into discrete molecules, and this is also the creator of the molecular diversity of the chemical matter (for further details, see the Retouches at the end of the lecture). 

Finite element modeling is a technique whereby a material continuum is divided into a number of patches, or finite elements, and the appropriate engineering theory is applied to solve a variety of problems. The initial (and probably still dominant) use of finite element modeling was for the solution of structural engineering problems. The technique is currently being applied by a number of companies and research institutions in the design of plastic products. CAD/CAM systems provide the means to create a mesh of finite elements directly from a product model database, by automatic and semiautomatic means. 

Greek philosophers viewed the physical world as matter organized in the form of bodies having length, breadth, and depth that could act and be acted upon. They also believed that these bodies made up a material continuum unpunctuated by voids. Within such a universe, they speculated about the creation and destruction of bodies, their causes, the essence they consisted of, and the purpose they existed for. Surfaces did not fit easily into these ancient pictures of the world, even those painted by the atomists, who admitted to the existence of voids. The problem of defining the boundary or limit of a body or between two adjacent bodies led Aristotle (fourth century BC) and others to deny that a surface has any substance. Given Aristotle s dominance in ancient philosophy, his view of surfaces persisted for many centuries, and may have delayed serious theoretical speculation about the nature of solid surfaces [2]. 

Here, C denotes the configuration tensor and D describes the rate-of-strain tensor of the material continuum. The dimensionless anisotropy factor, cc, characterizes the anisotropic character of the particle mobility. It is easy to show that a attains values between zero and one. The limiting case a = 0 corresponds to the isotropic motion and ultimately leads to an upper converted Maxwell material. In order to derive a deformation-dependent constitutive equation, a Hookean law connecting the tensor of external stresses S, the configuration tensor C, and the shear modulus G was suggested ... 

Application of the exact continuum analysis of dispersion forces requires significant calculations and the knowledge of the frequency spectmm of the material dielectric response over wavelengths X = 2irc/j/ around 10-10 nm. Because of these complications, it is common to assume that a primary absorption peak at one frequency in the ultraviolet, j/uv. dominates the dielectric spectrum of most materials. This leads to an expression for the dielectric response... 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

In this chapter we shall consider four important problems in molecular n iudelling. First, v discuss the problem of calculating free energies. We then consider continuum solve models, which enable the effects of the solvent to be incorporated into a calculation witho requiring the solvent molecules to be represented explicitly. Third, we shall consider the simi lation of chemical reactions, including the important technique of ab initio molecular dynamic Finally, we consider how to study the nature of defects in solid-state materials. 

One of the simplest ways to model polymers is as a continuum with various properties. These types of calculations are usually done by engineers for determining the stress and strain on an object made of that material. This is usually a numerical finite element or finite difference calculation, a subject that will not be discussed further in this book. 

Fracture mechanics is now quite weU estabHshed for metals, and a number of ASTM standards have been defined (4—6). For other materials, standardization efforts are underway (7,8). The techniques and procedures are being adapted from the metals Hterature. The concepts are appHcable to any material, provided the stmcture of the material can be treated as a continuum relative to the size-scale of the primary crack. There are many textbooks on the subject covering the appHcation of fracture mechanics to metals, polymers, and composites (9—15) (see Composite materials). 

Contact Drying. Contact drying occurs when wet material contacts a warm surface in an indirect-heat dryer (15—18). A sphere resting on a flat heated surface is a simple model. The heat-transfer mechanisms across the gap between the surface and the sphere are conduction and radiation. Conduction heat transfer is calculated, approximately, by recognizing that the effective conductivity of a gas approaches 0, as the gap width approaches 0. The gas is no longer a continuum and the rarified gas effect is accounted for in a formula that also defines the conduction heat-transfer coefficient ... 

Kinematical relations in large deformations are given here for reference. Most of the material is well known, and may be extracted or deduced from the comprehensive expositions of Truesdell and Toupin [19], Truesdell and Noll [20], or other texts in continuum mechanics, where further details may be found. 

Underlying all continuum and mesoscale descriptions of shock-wave compression of solids is the microscale. Physical processes on the microscale control observed dynamic material behavior in subtle ways sometimes in ways that do not fit nicely with simple preconceived macroscale ideas. The repeated cycle of experiment and theory slowly reveals the micromechanical nature of the shock-compression process. 

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

Another chapter deals with the physical mechanisms of deformation on a microscopic scale and the development of micromechanical theories to describe the continuum response of shocked materials. These methods have been an important part of the theoretical tools of shock compression for the past 25 years. Although it is extremely difficult to correlate atomistic behaviors to continuum response, considerable progress has been made in this area. The chapter on micromechanical deformation lays out the basic approaches of micromechanical theories and provides examples for several important problems. 

Example Approximate calculation of the hardness of solids. This concept of shear yielding - where we ignore the details of the grains in our polycrystal and treat the material as a continuum - is useful in many respects. For example, we can use it to calculate the loads that would make our material yield for all sorts of quite complicated geometries. 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

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